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A356844
Numbers k such that the k-th composition in standard order contains at least one 1. Numbers that are odd or whose binary expansion contains at least two adjacent 1's.
9
1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87
OFFSET
1,2
COMMENTS
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
FORMULA
Union of A005408 and A004780.
EXAMPLE
The terms, binary expansions, and standard compositions:
1: 1 (1)
3: 11 (1,1)
5: 101 (2,1)
6: 110 (1,2)
7: 111 (1,1,1)
9: 1001 (3,1)
11: 1011 (2,1,1)
12: 1100 (1,3)
13: 1101 (1,2,1)
14: 1110 (1,1,2)
15: 1111 (1,1,1,1)
17: 10001 (4,1)
19: 10011 (3,1,1)
21: 10101 (2,2,1)
22: 10110 (2,1,2)
23: 10111 (2,1,1,1)
24: 11000 (1,4)
25: 11001 (1,3,1)
26: 11010 (1,2,2)
27: 11011 (1,2,1,1)
28: 11100 (1,1,3)
29: 11101 (1,1,2,1)
30: 11110 (1,1,1,2)
31: 11111 (1,1,1,1,1)
MATHEMATICA
Select[Range[0, 100], OddQ[#]||MatchQ[IntegerDigits[#, 2], {___, 1, 1, ___}]&]
CROSSREFS
See link for sequences related to standard compositions.
The case beginning with 1 is A004760, complement A004754.
The complement is A022340.
These compositions are counted by A099036, complement A212804.
The case covering an initial interval is A333217.
The gapless but non-initial version is A356843, unordered A356845.
Sequence in context: A333227 A285510 A174415 * A103826 A361386 A361786
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 02 2022
STATUS
approved